Complete Convergence and Weak Law of Large Numbers for ρ-Mixing Sequences of Random Variables

  ; 1 X i  i is a strictly stationary Gaussian sequence which has a bounded positive every t, then 1 1 1 m M        ; 1 n X n  . Thus, is a  -mixing sequence.  -mixing is similar to  -mixing, but both are quite different.   k  is defined by (1.1) with index sets restricted to subsets S of   1,n T and subsets of   , , , n k n k N   . On the other hand,   -mixing sequence assume condition ,but   0 k    k N -mixing sequence assume condition that there exists  such that   1 k    , from this point of view,  -mixing is weaker than  -mixing. A number of writers have studied  -mixing sequences of random variables and a series of useful results have been established. We refer to [2] for the central limit theorem [1,3], for moment inequalities and the strong law of large numbers [4-9], for almost sure convergence, and [10] for maximal inequalities and the invariance principle. When these are compared with the corresponding results for sequences of independent random variables, there still remains much to be desired. The main purpose of this paper is to study the complete convergence and weak law of large numbers of partial sums of  -mixing sequences of random variables and try to obtain some new results. We establish the